(Because it may be relevant this year, I'm mostly listing things in
alphabetical order instead of any implied quality or preference order.)

Standouts:

Fullmetal Alchemist: People argue about whether this or Brotherhood
is the better FMA experience, but I hope that we can all agree that
this is excellent regardless of which is the better version. The anime
original ending doesn't bother me in the least and the journey there is
full of excellent twists and turns and any number of great characters.
There is so much good about this show that I have no real words for it.

This is the best shonen action show that I've seen. (Shonen action
shows are not the same thing as shonen fighting shows.)

Gunslinger Girl: On one level this is an exciting action story with a
thin veneer of plausibility, and on that level it will not let you
down although it will periodically disturb you, sometimes deeply.
At another level this is a character story about a bunch of extremely
broken people (some of them obviously so and some not). The ending is
quiet and beautiful and a gut punch in its own way.

(I was never able to take the followup series seriously. Read the
manga instead.)

Twin Spica: This is an excellent adaptation of (part of) what I
understand is an excellent manga. It features great characters,
including the protagonist, good writing, and so on, all wrapped up
in an affecting, heartfelt storyline, and the conclusion to the show
warmed the cold cockles of my heart as few other endings ever have.

(It's 20 episodes. That won't take long. Do check it out. Yes, this is
another underappreciated series that I'm trying to
sell you on.)

Dokkoida?!: You probably haven't heard of this but it's a quite
charming and amusing show. It's the sort of comedy that makes me
smile reliably (which is not common) and the characters are generally
touching, especially once you get to know them.

Godannar: I've never been able to decide if this show is a straight up
serious take on the Go Nagai part of the giant robot genre or a
straight-faced affectionate parody of it. Whichever it is, it's
epic. It has any number of plot twists, a bunch of cool characters
(including secondary characters like the mechanics), and GIANT COMBINING
ROBOTS in the Go Nagai tradition. Oh, and lots of fanservice of an
old-fashioned style (that goes with the Go Nagai tradition, of course).

Narutaru (aka Shadow Star): This starts out as a pleasant bright
series where a girl winds up with an alien pet and then things go
all to hell. It's basically all the fault of people. The results are
wrenching but memorable. Note that this is part of a manga so you
won't get any real answers in the show (but by the end you probably
won't care about that).

Stellvia: I have fond memories of watching this but I have to admit
that many of the actual details of what happened in the show have
faded by now. I'd probably enjoy a rewatch and I may do that someday.
I remember the romance as one of the better ones that I've seen.

Honorable mentions:

Chrono Crusade (sometimes aka Chrno Crusade): I'll be honest; this
is memorable in large part because of how it had a relative downer
of an ending. But I remember it as well executed on the way there,
with interesting characters (especially the protagonist).

Full Metal Panic? Fumoffu: This is okay, but I like the serious
Full Metal Panic better than the full bore pratfalls comedy version.

Last Exile: Scamp's review has made me look back
on this with more favour than I did at the time, partly by giving
me a more optimistic take on what the show did to Lavie. Parts of it
are great, parts of it I would have to rewatch to perhaps reassess,
and parts of the ending are well over the top.
Please note that the opening fleet combat segments are supposed to
be absurd and stupid and ritualized and ineffectual. That's the point.
The Guild wants them that way and what the Guild wants, it gets.

(The resemblance to the stupider parts of World War I warfare is
probably deliberate.)

Read Or Die the TV: Unsurprisingly the much longer TV series is much
more character focused and has much less action (and it's lower-powered)
than the OVA trilogy. It wasn't bad, though. Anita steals the show.

Scrapped Princess: Bagpipes in the opening, antagonists named after
firearms, and an interesting explanation for everything that was going
on. The show was not flawless and fumbles the execution at a few points.

Tsukihime: Apparently any number of people will assure you that this
anime doesn't exist. I found it a perfectly fine show with a number of
interesting concepts and interesting characters.

Yami to Boushi to Hon no Tabibito: See this Secret Santa review
for a pretty good summary of the show. When I watched it I thought it
was going to be deeper than it turned out to be and the ending is not
the greatest, but the bits in the middle are genuinely fun and crazy.

I've seen a number of other shows from 2003 that don't make these lists
for one reason or another.

Things I want to watch or that I know I should watch category:

The Big O (second season): I loved the first season but the second
season dropped into the void of disappearing Toronto anime clubs.

Kino's Journey: I saw the first few episodes but no more for some
reason. The few episodes I've seen have been enough to show me that
I want to watch more.

Tokyo Godfathers

I saw a fair amount of Gad Guard at the time and kind of want to
finish it off someday. It was a stylish show that took a significant
plot swerve just where I stopped being able to see more.